cold engine with high temperature code

Do you have an OBD2 scanner that you can use to check for trouble codes?

Based on your symptoms, there can be a short in the engine coolant temperature sensor circuit OR the engine coolant temperature sensor could be bad.

EDIT:
There is a second temperature sensor for the exhaust heat recovery system that can also turn on the high coolant temperature warning light. So, there could be a short in that circuit OR that temperature sensor could be bad.

I would first look for a short in both circuits because the temperature sensors very rarely go bad.